The SSM Procedure

Regression Variable Specification in Multivariate Models

Suppose that a regression variable in a multivariate model affects two or more response variables. For example, suppose that response variables y1 and y2 depend on a regression variable x. This dependence can be categorized as one of two types:

In the more common case, the regression coefficient of x for y1 and the regression coefficient of x for y2 are different. The relationship can be described as follows:

$\begin{eqnarray*} y1 & = & \beta _{1} x + \text {other terms} \nonumber \\ y2 & = & \beta _{2} x + \text {other terms} \nonumber \end{eqnarray*}$

In the SSM procedure you can specify this type of relationship in two equivalent ways:
- You can specify the variable x in the MODEL statement for y1 and specify the variable x_copy (a copy of x) in the MODEL statement for y2 as follows:
```
x_copy = x;            /* create a copy of x */
model y1 = x ...;
model y2 = x_copy ...;
```
- You can specify the variable x in MODEL statements for both y1 and y2 as follows:
```
model y1 = x ...;
model y2 = x ...;
```
  This specification avoids creating x_copy.
Of these two alternate ways, the first is preferred because x and x_copy can then be unambiguously used in an EVAL statement to refer to the terms $\beta _{1} x$ and $\beta _{2} x$ , respectively.
In the less common case, y1 and y2 share a common regression coefficient.The relationship can be described as follows:

$\begin{eqnarray*} y1 & = & \beta x + \text {other terms} \nonumber \\ y2 & = & \beta x + \text {other terms} \nonumber \end{eqnarray*}$

You can specify this type of relationship by placing the regression coefficient in the model state vector as follows:
```
state beta(1) T(I) A1(1) ;       /* beta is a constant state */
comp xeffect = beta*(x) ;
model y1 =  xeffect ...;
model y2 =  xeffect ...;
```
Here the STATE statement defines beta as a one-dimensional, time-invariant constant (because the transition matrix is identity, the disturbance covariance is 0 and the initial state is diffuse). Next, the COMP statement defines xeffect as the product between beta and the variable x. Subsequently, both y1 and y2 use xeffect in their respective MODEL statements.